Hyatt is offering 2,500 American AAdvantage bonus miles per Hyatt stay through October 15, which is in addition to the normal 500 miles, for a total of 3,000 miles per stay. Registration is required here. You can earn a maximum of 60,000 points through this promotion, though I doubt many of us have more than 20 stays we’d want to credit to miles during that period.
In addition to that, you can earn 1,000 bonus miles per night through select airline programs by calling Gold Passport and registering for promotion TP30K2010, which runs through September 30. And it seems like this promotion can be combined with the American promotion, for a minimum of 4,000 American miles per stay.
And since the question often comes up, you have to choose between points OR miles on Hyatt stays. The only points you earn if you credit a stay to miles would be the Diamond welcome amenity or any points you get as compensation (like not getting preferred bed type as Diamond, closed lounge for Diamond, etc.).
(Tip of the hat to Gary)